THE EGS5 CODE SYSTEM

integrating over azimuth, and changing from an angle variable to an energy variable, we obtain the
following form of the differential cross section, which is used in PEGS:
where
d˘Σ Annih (Ĕ0)
d˘k
= S 1 (k ′ ) + S 1 (A − k ′ ) (2.222)
Ĕ 0 = energy of incident positron (MeV), (2.223)
˘k = energy of secondary photon of lower energy (MeV),
γ = Ĕ0/m, (2.224)
A = γ + 1 = (available energy)/m, (2.225)
T 0 ′ = γ − 1 = (kinetic energy)/m, (2.226)
√ √
p ′ 0 = ˘p 0 /m = γ 2 − 1 = AT 0 ′ (2.227)
k ′ = ˘k/m, (2.228)
S 1 (x) = C 1 [−1 + (C 2 − 1/x)/x], (2.229)
C 1 = X 0nπr 2 0
AT ′ 0 m , (2.230)
C 2 = A + 2γ/A . (2.231)
We see that Equation 2.222 satisfies, in a manifest way, the symmetry under exchange of the
annihilation photons.
Integrating Equation 2.222 between ˘k 1 and ˘k 2 we obtain
where
∫ ˘k2
˘k 1
d˘Σ Annih
d˘k
∫
S 2 (x) = m
d˘k = S 2 (k ′ 2 ) − S 2(k ′ 1 ) + S 2(A − k ′ 1 ) − S 2(A − k ′ 2 ) (2.232)
k ′ i = ˘k i /m, i = 1, 2, (2.233)
S 1 (y)dy = m C 1 [−x + C 2 ln x + 1/x] . (2.234)
For the total annihilation cross section we use Heitler’s formula with appropriate changes to
take into account units and notation, and we have
˘Σ Annih (Ĕ0) = X [
0nπr0
2 γ 2 √
+ 4γ + 1
γ + 1 γ 2 ln(γ + γ
− 1
2 − 1) − √ γ + 3
]
. (2.235)
γ 2 − 1
PEGS functions ANIHDM, ANIHRM, and ANIHTM evaluate Equations 2.222, 2.232, and 2.235, respectively.
In computing the energies of the secondary photons, we sample the parameter E, which is
defined by
E = k′
A =
˘k
Ĕ 0 + m . (2.236)
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